Here’s a fun story: During my first semester at JMU, I had gotten it in my head to learn algebraic topology. This obviously did not quite work out. However, one thing I was able to grasp onto was Knot Theory! I watched some lectures and tried to solve a few of the exercises in computing the Alexander-Conway polynomial of a few knots, but eventually realized that I had no idea if I was doing it correctly or not.

So I looked at JMU’s faculty list and searched for Knot Theory, finding Dr. Laura Taalman. I went to her office, said

“Hi, my name is Jonathan, can I ask you something about Knot Theory?”

She replied,

“Well that sounds like more fun than anything I’m doing right now, so sure!”

We proceeded to do an awesome project involving knot invariants and 3D printing, which you can find more info about in this blog. To make a long story short, Laura left for a few years to do some cool work with MakerBot and MoMath, but returned for my senior year and we figured we had to do a project together.

I finally got the algebraic topology class I desired so much last semester. I was in Moscow, Russia from February 2 to May 22 studying mathematics and Russian at the Independent University of Moscow and the Higher School of Economics. This was an amazing experience, which at the risk of an already-long blog post, I will not expand on at the moment!

So here’s the plan for our project: Use 3D printing to create a series of models demonstrating homotopies between two objects.

A homotopy between two objects (specifically two topological spaces) is a continuous deformation of one into the other. These can sometimes be hard to visualize. The classic example (and basis for a *hilarious* topology joke) is of the donut and the coffee mug. These are homotopy equivalent (that is, there exists a continuous deformation from one into the other), which we can see with this helpful gif:

In fact, we found out that Henry Segerman has already printed a series of 3D models to demonstrate this homotopy, which can be seen here. We want do a similar project with more complicated homotopies. Though we will certainly be looking to 3D print some nice physical deformations between two objects, something we really want to do is take a hard-to-visualize formal homotopy and bring it to life!

To finish this post, I want to leave you with a video. We hope to animate our transformations using a Zoetrope! EDIT: Here’s a video demo!

I’ll have more information in future posts!

Thanks for reading,

Jonathan

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